MinT - Best Known (180−84, 180, s)-Nets in Base 2

Best Known (180−84, 180, s)-Nets in Base 2

(180−84, 180, 54)-Net over F₂ — Constructive and digital

Digital (96, 180, 54)-net over F₂, using

t-expansion [i] based on digital (95, 180, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 5 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(180−84, 180, 65)-Net over F₂ — Digital

Digital (96, 180, 65)-net over F₂, using

t-expansion [i] based on digital (95, 180, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(180−84, 180, 224)-Net over F₂ — Upper bound on s (digital)

There is no digital (96, 180, 225)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁰, 225, F₂, 84) (dual of [225, 45, 85]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(2¹⁷⁹, 209, F₂, 84) (dual of [209, 30, 85]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁸⁰, 210, F₂, 85) (dual of [210, 30, 86]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(2⁴⁵, 225, S₂, 16), but
    - discarding factors would yield OA(2⁴⁵, 189, S₂, 16), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 36 338770 420561 > 2⁴⁵ [i]

(180−84, 180, 264)-Net in Base 2 — Upper bound on s

There is no (96, 180, 265)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 621440 756384 794269 353417 929469 538729 688210 476425 537376 > 2¹⁸⁰ [i]